# How many counterterms does QED have?

I have read the statement that QED has four counterterms to cancel divergences. However, I have learnt that there are only three counterterms (vertex, electron propagator, photon propagator), which is consistent with this PDF$$^1$$ (42KB).

But then, the last page of this PDF$$^2$$ (229KB) states that there are five kinds of diverging skeleton diagrams in QED, one of which vanishes (the triangle).

Is there a fourth counterterm or did the original statement actually refer not to counterterms, but to diverging diagrams?

$$^1$$ "QED Feynman Rules in the Counterterm Perturbation Theory" by Vadim Kaplunovsky (University of Texas)

$$^2$$ "Renormalization in QED" (Imperial College London)

## 1 Answer

Whether there are three or four counterterms just depends on how you count. There are only three divergent diagrams, because of two facts: First, diagrams with only odd numbers of external photons attached vanish; this can be most easily proven using invariance under C (charge conjugation). Second the four-photon diagram, although it is superficially divergent, is actually finite, because of gauge invariance. This diagram leads to the finite Euler-Heisenberg correction to the action.

How you get four counterterms from three divergent diagrams is by noticing that the electron self-energy contains two divergent pieces with different Lorentz structures. There is a kinetic term, related to the fermion field strength renormalization, and also a mass renormalization term. If you count these as representing two separate counterterms (which may or may not be useful, depending on what you are trying to do), they—along with the vertex correction counterterm and the photon field strength—give four total.

• Makes perfect sense. In the first PDF I've linked, there are only three diagrams, but they depend on four parameters: $\delta_1$, $\delta_2$, $\delta_m$ and $\delta_3$. – Stephan Nov 11 '18 at 10:12